Following the approximant path and moving back towards quasiperiodic models, we approach an irrational magnetic flux by taking an appropriate rational limit with \(p,q \rightarrow \infty \) . In this limit, the
spectrum becomes a Cantor set fractal [1]}. At each approximant generation, new gaps can open, exhibiting quantized Chern numbers. It is important to note that depending on the model, the gaps can close and reopen under some perturbation that does not break the unit cell, see Fig. REF . In such a case, the Diophantine equation still holds and the corresponding Chern numbers can change by a multiple of \(q\)  modulo \(q\) . Nevertheless, at later approximants, due to the changed denominator \(q_{n+l}> q_n\) , such a exchange of Chern numbers is no longer possible and the gap may close but without a change to the Chern number. Crucially, in order to change a topological index of the gap, a phase transition must occur and the gap must close, i.e., there is no way to deform between the two distinct topological states without the phase transition. However, picking a specific deformation [as for example done in the IAAF model (REF )], gaps can close and reopen, but it does not necessitate a change in the topological index. Combining the two arguments in this paragraph, alongside the fact that all gaps are known to be open in both limits of the IAAF defomration, we could color the gaps of the Fibonacci limit of the model [2]}.
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